EMBEDDING OPERATORS INTO STRONGLY CONTINUOUS ...

EMBEDDING OPERATORS INTO C 0 -SEMIGROUPS 4
Question 3.4. For which (classes of) operators the necessary condition given in Theorem 3.1
is also sufficient for embeddability?
Note that the condition given in Theorem 3.1 is not sufficient for embeddability in general. For
example, Halmos, Lumer an Schäffer [14] constructed a class of invertible operators on Hilbert
spaces which have no square root. Later, Deckard and Pearcy [4] showed that such operators
exist on every infinite-dimensional Hilbert space.
4. Embedding isometries on Hilbert spaces
In this section we characterise the embedding property for isometries, co-isometries and projections
on Hilbert spaces. Note that the spectrum of a non-invertible isometry is the unit disc
(see Conway [3, Exercise VII.6.7] for the Banach space case or Theorem 4.2 below for isometries
on Hilbert spaces), and hence the spectral calculus method is not applicable.
We first recall that unitary operators are embeddable.
Proposition 4.1. Every unitary operator U on a Hilbert space H can be embedded into a unitary
C 0 -group with bounded generator.
Proof. By the spectral theorem, see e.g. Halmos [13], we can assume that U has a cyclic vector,
H = L 2 (Γ, µ) for the unit circle Γ and some Borel measure µ and
Define now
(Uf)(z) = zf(z), z ∈ Γ, f ∈ L 2 (Γ, µ).
(U(t)f)(e iϕ ) := e itϕ f(e iϕ ), f ∈ L 2 (Γ, µ),
for ϕ ∈ [0, 2π) and t ∈ R. The operators U(t) form a unitary C 0 -group which is even uniformly
continuous and satisfies U(1) = U.
□
To treat the general case we need the following structure theorem for isometries on Hilbert
spaces.
Theorem 4.2. (Wold decomposition, see Sz.-Nagy, Foiaş [20, Theorem 1.1].) Let V be an
isometry on a Hilbert space H. Then H can be decomposed into an orthogonal sum H = H 0 ⊕H 1
of V -invariant subspaces such that the restriction of V on H 0 is unitary and the restriction of
V on H 1 is a unilateral shift. More precisely, for Y := (rg V ) ⊥ ⊂ H 1 one has V n Y ⊥ V m Y for
all n ≠ m ∈ N 0 and H 1 = ⊕ ∞
n=0 V n Y .
So, by Wold’s decomposition and Proposition 4.1, the problem of embedding an isometry
reduces to embedding the right shift on the space l 2 (N, Y ) for some Hilbert space Y . The
following result shows when this can be achieved.
Proposition 4.3. Let T be the right shift on l 2 (N, Y ) for an infinite-dimensional Hilbert space
Y . Then T can be embedded into an isometric C 0 -semigroup.
Proof. Let T be the right shift on l 2 (N, Y ), i.e.,
T(x 1 , x 2 , x 3 , . . .) := (0, x 1 , x 2 , . . .).
By general Hilbert space theory, Y is unitarily isomorphic to the space L 2 ([0, 1], Y ), hence there
is a unitary operator J : l 2 (N, Y ) → l 2 (N, L 2 ([0, 1], Y )) such that JTJ −1 is again the right shift